1. ## Mapping

What does it mean to map something. Specifically, I'm trying to understand this proof of the fundamental theorem of algebra and it is totally incomprehensible. In the first paragraph it says "Now we will think of the polynomial function as mapping the complex plane to itself." Maybe if I can understand this sentence I'll be able to handle it from there.

2. Originally Posted by zg12
What does it mean to map something. Specifically, I'm trying to understand this proof of the fundamental theorem of algebra and it is totally incomprehensible. In the first paragraph it says "Now we will think of the polynomial function as mapping the complex plane to itself." Maybe if I can understand this sentence I'll be able to handle it from there.
A function receives as input an element of some set A and returns as output an element of some set B. We say f maps A to B. (A is the domain and B is the codomain.)

For example, consider a function f that takes any real number and returns 0. This function maps $\displaystyle \,\mathbb{R}$ to {0}.

Common notation is $\displaystyle \,f:\mathbb{R}\to\{0\}$

We can also use the word map for individual elements. Thus, for the above function we have f maps 3 to 0, f maps 6.5 to 0, etc.

Input/output is a common computer science analogy; the usual way to define a function is by way of a subset of the Cartesian product $\displaystyle A\times B$ with every element of A paired with (mapped to) exactly one element of B.